# Full text of "Analytical Mechanics"

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```324                     ANALYTICAL MECHANICS
249. Logarithmic Decrement. — The logarii hm of the ratio of two consecutive amplitudes is constant and is called the logarithmic decrement of the motion. The amplitudes occur whenever the relation
CO*
tan (at) = — a
is satisfied. Let the first amplitude occur at the instant t = h; then since the period of the tangent is TT, the times of the succeeding amplitudes are given by
tan (wf) = tan (wti + HT) , u                               /     / _L 7l7r
or by                                   t  =  Cl + ~~ »
CO
where n is a positive integer. Hence, denoting the logarithmic decrement by X and the nth amplitude by anj we have
X = log —2-        (by definition)
a» + 2
-        ^i        -    Sin^wfi ±
"                   "         "'""
A6""flvl   " ...... « v sin [wfe + (n + 2)
a(,rf^)
!       6         V         w/
= log - r-TTTT
2ir
~~~ —
0)
= ~P.                                              (XXV)
Therefore if I is known k" may be determined from observations of P and a.
* Obtained by setting^- » 0.```